1Department of Mathematics, Shandong University of Science and Technology, Qingdao 266590, China2Department of Statistics and Finance, Shandong University of Science and Technology, Qingdao 266590, China

Abstract

By establishing a comparison result and using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the
existence of solutions for nonlinear fractional differential systems with coupled four-point boundary value problems.

1. Introduction

This paper discusses the coupled four-point boundary value problems
where and are continuous, , with , and denotes the Caputo fractional derivative of with defined by
is the Riemann-Liouville fractional integral of order ; see [1–4].

It is well known that

Fractional differential equation’s modeling capabilities in physics, chemistry, economics, and other fields, over the last few decades, have resulted in the rapid development of the theory of fractional differential equations; we refer the reader to the books [1–4]. On the other hand, the study of systems involving coupled boundary value problems is also important as such systems occur in the study of reaction-diffusion equations and Sturm-Liouville problems, for example, [5–16]. In [17–25], using the upper and lower solutions method and the monotone iterative method, the authors considered the existence of solutions of initial value problems and boundary value problems for fractional differential equations. But, as far as we know, there have been few papers which have considered the existence of solutions of (1) by means of the monotone iterative method.

Motivated by the above papers, in this paper, we will investigate the existence of a solution of problem (1) by means of the upper and lower solutions method and the monotone iterative method. The novelty of this paper is that Caputo-type fractional differential systems involve two different fractional derivatives and and that the nonlinear terms , in the systems (1) involve unknown functions and .

In the following, we denote

2. Preliminaries and Lemmas

In this section, we introduce the definition of the lower and upper solutions and present some existence and uniqueness results for linear problems together with comparison results for differential systems (1) which will be needed in the next section.

Throughout this paper, we always assume that the following condition is satisfied:

.

Definition 1. is called a lower system of solutions of differential system (1) if
Analogously, is called an upper system of solutions if it satisfies the reversed inequalities.

If and , , we say that and are ordered lower and upper system of solutions of (1). In what follows, we assume that and are ordered lower and upper system of solutions of (1) and define the sector
where the vectorial inequalities mean that the same inequalities hold between their corresponding components.

Proof. Suppose the contrary. By Lemma 2, We consider the following three possible cases.Case 1. Consider and . By Lemma 2, , . Then which contradicts . Case 2. Consider and . By Lemma 2, , . Then which contradicts . Case 3. Consider and . By Lemma 2, we have and . We only prove that . If not, has a local positive maximum at some such that . Then, by Theorem 2.1 in [21], we have the fact that the Caputo derivative of the function is nonpositive at the point . Thus,
which is a contradiction. Furthermore, considering the boundary condition , there exists such that
A similar proof, for , gives us that there exists such that
It follows from (10) and (11) that
which implies that , a contradiction. Hence , , .

Corollary 4. Let be given. Assume that satisfy
Then , .

Lemma 5. Let , , then the linear differential system with coupled four-point boundary value problem
has integral representation
where

Proof. It follows from [21] that (14) is equivalent to the system of integral equations
By coupled four-point boundary value conditions of problem (14), we have
After simple computation, we get
Substituting (20) into (18) and (21) into (19), respectively, we obtain the desired results.

Now we enunciate the following existence and uniqueness results for differential system:
where , .

Proof. Indeed, by Lemma 5, differential system (22) is equivalent to the operator equation
where
We apply the Fredholm theorem to find the unique solution of differential system (22). By using standard arguments, we can easily show that is linear completely continuous. Also, by Corollary 4, the operator equation has only the zero solution. Thus, for given , operator equation (23) has a unique solution in , by the Fredholm theorem. This ends the proof.

3. Main Results

In this section, we prove the existence of extremal solutions of differential system (1).

Theorem 7. Assume that , . Let and be ordered lower and upper system of solutions of (1). In addition, we assume that is nondecreasing in and there exists such that
where , ; is nondecreasing in and there exists such that
where , .

Proof. Let us define two sequences by relations
for . Note that are well defined, by Lemma 6. First, we show that
Let , . This and the assumption that is a lower system of solutions of (1) yield
Hence, and , , by Lemma 3. By a similar way, we can show that and , . Now we put , . Hence, in view of assumptions , , we have
This and Lemma 3 prove that , , so, relation (28) holds.Now we show that is a lower system of solution of problem (1). Note that
by assumptions , . It proves that is a lower system of solution of (1). Similarly, we can prove that is an upper system of solution of problem (1).By mathematical induction we can show that
for and . Employing standard arguments we see that the sequences converge to their limit functions , , respectively. Indeed, and are solutions of problem (1) and on .We need to show now that and are extremal solutions of problem (1) in the segment . To prove it, we assume that is another solution of problem (1) and , for some positive integer . Put , . Hence, in view of assumptions , , we have
Hence, , , by Lemma 3. By a similar way, we can show that , . So by induction, , on for all . Taking the limit as , we conclude , . That is, and are extremal systems of solutions of (1) in .

4. Example

Consider the following problems:
Obviously,
Take , ; then
It shows that and are lower and upper systems of solutions of (34).

On the other hand, it is easy to verify that conditions , hold for and .

By Theorem 7, problem (34) has an extremal system of solutions and , which can be obtained by taking limits from some iterative sequences.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. The Project is supported by the National Natural Science Foundation of China (11371221, 61304074), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province, the Postdoctoral Science Foundation of Shandong Province (201303074), and Foundation of SDUST.